1. A topology-based approach towards
understanding mixing in high-speed flows
Sawan Suman
Post-doc
Turbulence Research Group
Texas A&M University

2. Application Navier-Stokes Equations DNS 7-eqn. RANS ARSM reduction
Body force effects
DNS
Linear Theories: RDT
7-eqn. RANS
Spectral and non- linear theories
Realizability, Consistency
ARSM reduction
Mixing in high speed environment
2-eqn. RANS
Averaging Invariance
2-eqn. PANS
Near-wall treatment, limiters, realizability correction
Numerical methods and grid issues
DNS
LES
RANS
Application

3. Introduction
Enhanced scalar mixing: essential in ramjet/scramjet combustors
Compressibility reduces KE: reduces turbulent mixing
Understanding/modeling/improvement at two levels:
(a) Macro stirring:
(b) Micro mixing: Scalar dissipation must be enhanced
Requires understanding of small-scale structures: scalar- and velocity-gradients
Aims:
-to understand the role played by the structure of velocity gradient in shaping the behaviour of this term?
-to quantify the mixing capability of various possible structures of velocity gradient

4. Structure of velocity-grad field: Topology
Local flow-field topology: Visual, intuitive and physically sensible way to study
velocity-gradient structure
Topology= Local streamline pattern within a fluid element/Exact deformation pattern of a fluid
element
Pattern of streamlines = Nature of eigen values of the tensor
Reference:
Strain-dominated topology, real eigenvalues Rotation-dominated topology, complex eigenvalues
Stable-node Stable-focus

5. Introduction 3-D flows have more complex topologies
Unstable node/saddle/saddle (UNSS), Stable focus stretching (SFS), etc.
Compressible flows have more possible topologies compared to incomp. flows
(Chong & Perry, 1990, POF, Suman & Girimaji, 2010, JoT)
Unstable focus stretching (UFS), Stable focus compressing (SFC), etc
Which topologies in compressible turbulence are more efficient in mixing?
CFD analysis and design can aim to maximize the population of efficient topologies

6. Velocity-gradients & mixing
How can velocity gradient maximize production of scalar dissipation?
Normalized evolution equation: time normalized by velocity gradient magnitude
Decomposition of velocity gradient: strain-rate and dilatation and rotation
Simpler form of evolution equation:
What do we know about the “incompressible” mixing?
Velocity gradient tensor
“Incompressible” mixing
“compressible” mixing

7. Known incomp. behaviour
Scalar dissipation maximum when scalar grad. aligned with large, -ve strain- rate
Scalar gradient is found to be aligned with large negative strain-rate
Vorticity mis-aligns scalar grad., reduces dissipation
Ashurt et al. (POF,1987), Brethouwer et al, (JFM, 2003), O’Neill et al (Fluid Dynamics Research, 2004)
vorticity vector
Plane of
Scalar gradient

8. Mixing efficiency definitions
Definitions take into account the role of velocity field only, scalar field is not accounted for
Will check the validity of this approach
Will compute volume averaged values of efficiency in decaying turbulence

9. Incompressible Turbulence

10. Incompressible turbulence:
Stable node/Saddle/Saddle
Unstable node/Saddle/Saddle
Stable-focus Stretching
Unstable-focus Compressing
UNSS best mixer

11. Validation
Scatter plot of scalar dissipation in DNS of incompressible turbulence
Stable-focus Stretching
Stable node/Saddle/Saddle
Unstable-focus Compressing
Unstable node/Saddle/Saddle
Stable node/Saddle/Saddle
O’Neill & Soria (Fluid Dynamics Research, 2004)
DNS of scalar field shows UNSS has highest scalar dissipation
Our approach – despite being based on only velocity-field information – reaches the same conclusion

12. Compressible Turbulence
Using DNS results of compressible turbulence
Only velocity-field available
No scalar

13. Stable node/Stable node
Contracting fluid elements
Unstable node
/Saddle/Saddle
Stable node/Saddle/Saddle
Stable node/Stable node
/Stable node
Stable-focus compressing
Stable-focus Stretching
Unstable-focus Compressing
SN/SN/SN
SN/SN/SN (isotropic contraction) best mixer
All contraction topologies better mixers than the incompressible ones, dilatational shrinking favors mixing

14. Incompressible turbulence:

15. Action of velocity field on scalar field
Compressive strain pushes iso-scalar surfaces closer, increasing scalar dissipation in that direction only
Iso-scalar surfaces
Negative dilatation (volume contraction) amplifies this process in all directions
– possible only in compressible flows

16. Unstable node/Unstable node
Expanding fluid elements
Stable node/Saddle/Saddle
Unstable node
/Saddle/Saddle
Unstable node/Unstable node
/Unstable node
Stable-focus Stretching
Unstable-focus Compressing
Unstable-focus Stretching
UN/UN/UN
B (UNSS) best mixer, UN/UN/UN (isotropic expansion) worst mixer
All topologies less efficient than incompressible topologies; negative contribution from dilatation

17. Incompressible turbulence:

18. Isotropic contraction
Conclusions
A topology–based approach proposed to study the association of velocity field and scalar mixing
Method reproduces major conclusion from DNS of incomp. turbulence with scalar mixing
Preliminary predictions for compressible flows:
-Mixing efficiencies: Contracting > Incompressible > Expanding fluid elements
-SN/SN/SN: isotropic contraction is the best mixer in compressible turbulence
-UNSS best mixer in incompressible and expanding fluid elements
Future work:
-Needs further validation with DNS of canonical compressible flows with scalars
-Combustor design: maximize isotropic contraction
Isotropic contraction
Best mixers